Download MOLECULAR INTERACTIONS r0 r0

MOLECULAR INTERACTIONS
Attempt the following questions from recent finals papers.
NB: I’ve given you the complete question in each case. If parts of questions are repeated in later questions, just
refer back to your previous answer – don’t feel you need to write it all out again!
Q1. 1999 General Physical Chemistry I, Q9
a) Write brief notes to explain the physical origin of the following types of intermolecular interaction:
i) inductive
ii) dispersive
iii) repulsive
b) The Lennard-Jones potential for the interaction of two argon atoms may be written in the form
V(r) = 4ε  
r012 r06
 r   r  
where r0 = 342 pm and ε = 1.712 x 10-21 J.
i) Sketch the form of this potential and state which of the types of force described above contribute to
each term.
ii) Derive an expression for the force between the two argon atoms at an arbitrary distance r.
iii) Calculate the radial position of the minimum in the potential and show that the well depth is equal to ε.
iv) Assuming that in liquid argon the coordination number of each atom is 12, estimate the enthalpy of
vaporization of liquid argon at 87 K. Describe any additional assumptions made in your calculation and
compare your answer with the experimental value, ∆vapH = 6.28 kJ mol -1 .
Q2. 2000 General Physical Chemistry I, Q5
a) Van der Waals attractions occur between molecules with and without permanent dipoles. Describe qualitatively
the physical origins of the three principle types of attractive Van der Waals intereactions (i.e. those whose
energy varies with distance as r-6.)
b) Attractive van der Waals interactions between individual molecules are usually weak (≤ kT at room
temperature) and short range; interactions between colloidal particles are generally strong (>> kT) and long range.
Comment on the consequences for the stability of colloids.
c) Two polystyrene spheres with polarisability α per unit volume and radius R are separated by a distance D.
R
D
R
The attractive part of the pair potential for the two particles has the form
AR
U=12D
for D << R
where the Hamaker constant A = 1.0 x 10-19 J for polystyrene spheres in a vacuum.
Estimate the interaction energy between the two particles of radius 0.5 µm and separatoin 0.1 µm. Compare the
interaction energy to the thermal energy at 298 K.
Would the interaction between the two spheres be weaker or stronger in water than in a vacuum?
d) The two spheres are placed in a uniform electric field of strength E. Explain how this field gives rise to an
additional interaction between the two spheres, which can be either positive or negative depending on the
direction of the electric field.
Show that the repulsive interaction energy due to the field has a maximum value of
U=
[(4/3)πR3] 2 α 2E 2
4πε0(D+2R)3
Note: the interaction energy between two parallel point dipoles separated by a distance r in a vacuum is
µ2
U=
.
4πε0r3
Q3. 2000 General Physical Chemistry II, Q9
a) The Random Flight Model is the simplest model for describing the conformations of polymer chains in solution.
A polymer is treated as consisting of N segments, each of which is characterised by a vector lk of length l. In the
Random Flight Model, the orientations of these segments are random, i.e. <lj.lk > = 0, where j and k are two
different segments. The vector R connecting the two ends of the polymer is
N
R = ∑ lk
k=1
Show that the mean square distance between the two ends can be written
<R2> = <R.R> = N l2
Calculate the predicted value, in the Random Flight Model, of the root-mean-square end-to-end distance, <R2> 1/2,
for polyethylene oxide of molecular weight 106 u, taking the segment to be (CH2CH2O) with a molecular weight of
44 u and a length of 0.36 nm.
The value of <R2> 1/2 inferred from light scattering is 110 nm. Comment on the Random Flight Model in the light of
this value.
b) Man-made polymers have several features that distinguish their behaviour from that of small molecules, for
example, a high molecular weight, polydispersity, and a large contribution to the standard entropy from different
chain conformations. Explain how these factors play a role in THREE of the following observations on the
behaviour of polymers.
i) For most polymers, determination of the molecular weight by osmotic pressure measurements and by
light scattering give different values;
ii) Polymer solutions show large deviations from ideality;
iii) Rubber warms up if it is stretched adiabatically (for a metal spring, only a very small temperature
change is observed);
iv) Weakly adsorbed polymers stabilize colloids against coagulation;
v) The radius of gyration of polystyrene in cyclohexane increases with increasing temperature.
Q4. 2001 General Physical Chemistry I, Q9
a) Explain what is meant by the second virial coefficient of a gas.
b) The second virial coefficient B of a gas is related to the intermolecular potential U between two molecules at a
distance r by
∞
- U /k T
2
B = 2π ⌠
⌡ (1-e ) r dr
0
i) Show that for molecules that behave like hard spheres of diameter σ (U = ∞ when r < σ, U = 0 when r >
σ) the virial coefficient is given by
B=
2π σ3
3
ii) For real molecules there is also an attractive interaction. An approximate representation of the
interaction potential can be made using the square well potential shown below.
∞
σ
U
λσ
distance
ε
Show that the expression for the virial coefficient now becomes
B=
2π σ3 3
[λ + (1-λ3)eε/kT]
3
iii) The temperature at which N2 behaves like an ideal gas (the Boyle temperature) is 325 K. Given that λ
is 1.58 for N2 use the value of the Boyle temperature to calculate the depth ε of the square well
potential.
iv) Experimentally determined values of the second virial coefficient of N2 at lower temperatures are
given below.
T/K
B / cm3 mol-1
100
-163.6
125
-104.8
175
-50.1
250
-16.2
Using the values of ε and λ from iii) show that the square well potential accounts accurately for the data
in the table above. Use the data to calculate σ.
v) What other experimental information could be used to show that the square well potential is not an
accurate representation of the true intermolecular potential?
Q5. 2001 General Physical Chemistry II, Q7.
a) Sketch a diagram of the typical behaviour of the surface tension of an aqueous solution of a
surfactant over a concentration range from well below the critical micelle concentration to just above it.
b) The adsorption of a non-ionic surface active agent at the air surface of an aqueous solution obeys the
Gibbs equation
dγ = -Γ RT dlnc
The values of the surface tension for an aqueous solution of a soluble polymeric surfactant at
concentrations just below its critical micelle concentration (10-3 M) at 300 K are given below.
concentration / M
surface tension / 10-3 Nm-1
5x10-5
37.4
1x10-4
36.5
5x10-4
34.4
1x10-3
33.9
Use these data to calculate the surface excess of the surfactant and hence calculate the area per
molecule at the air/water interface.
c) The hydrophilic part of the polymeric surfactant in part b) consists of a number of ethylene oxide
(EO) segments. The area per molecule occupied by the small molecule surfactant C12H25(OC2H4)6OH just
below its critical micelle concentration is 0.55x10-18 m2.
Assuming that the area per EO segment is the same for the two surfactants in their saturated
monolayers, estimate the number of EO groups in the polymeric surfactant.